On algebras of double cosets of symmetric groups with respect to Young subgroups

We consider the subalgebra $\mathrm{\Delta }$ in the group algebra of the symmetric group $G=S_{n_1+\cdots +n_{\nu }}$ consisting of all functions invariant with respect to left and right shifts by elements of the Young subgroup $H:=S_{n_1}\times \cdots \times S_{n_{\nu }}$. We discuss structure constants of the algebra $\mathrm{\Delta }$ and construct an algebra with continuous parameters $n_1$ extrapolating algebras $\mathrm{\Delta }$, it can be also can be rewritten as an asymptotic algebra as $n_j\to \mathrm{\infty }$ (for fixed $\nu$). We show that there is a natural map from the Lie algebra of the group of pure braids to $\mathrm{\Delta }$ (and therefore this Lie algebra acts in spaces of multiplicities of the quasiregular representation of the group $G$ in functions on $G\mathrm{/}H$).